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Originally Posted by: Ber7 The parameterization by Draghilev method.The starting point is taken near the bifurcation point (0,0).
Just a very little modifications: using norme(J) and the SMath ability for handling undefined parameters. ContourTifoleum.sm (17kb) downloaded 12 time(s).Best regards. Alvaro.

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on 09/11/2018(UTC), on 09/11/2018(UTC)


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Thank you, Alvaro. The norme function simplifies the code and reduces the calculation time. Example of using norme when solving ODE LorenzPointsAreEquidistantc.sm (7kb) downloaded 13 time(s).

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Hi Ber. I don't remember to read nothing in the literature about equally spaced points in the numerical solution of the ode, except for the opposite: adaptive steps, but referring for the time variable, not the X,Y,Z solution points. About how you apparently get the same distance between solution points, i.e. sqrt(X^2+Y^2+Z^2), I guess that the background theory must to be in the Draghilev method and how the solution (X,Y,Z) is obtained from the differential equation. You have a very interesting point for investigate and publish about it. I try to investigate the relationship between the symbolic ode solution and the paramatrization, but the symbolic solutions are quite complicated, and I don't have simple examples. Apparently the distance between the points is 1 (guess can be easily proved because you divide by the norme the system), and this seems to provide more stable numerical solutions for the system (can be applied here Lyapunov's theorems?) Unfortunately in this example I introduce the factor 1/1000 for avoid numerical over max limit error for the case without norme. Best regards. Alvaro.




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Originally Posted by: Ber7 The parameterization by Draghilev method.The starting point is taken near the bifurcation point (0,0). Thanks Ber7, gorgeous For this particular Trifolium, once created it is easy to collect as many as desired to any scale corresponding to Draghilev 'a' The great tool here, is the bidirectional fmesh(f(x),x0,x1,mesh) Cheers ... Jean 2D Parametric Plot [Create Trifolium].sm (32kb) downloaded 11 time(s).




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Thank you Jean, I suggest a small change in the animation. TrifoliumAnim.sm (25kb) downloaded 11 time(s).Edited by user 11 November 2018 12:44:29(UTC)
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Originally Posted by: Ber7 I suggest a small change in the animation. Thanks Ber7 ... even more compacted. Trifolium:=stack(b1,b2,b3,b4) from inversing f4(x) <= f3(x). 0Anim Trifolium [Windmill Ber7].sm (26kb) downloaded 10 time(s).




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Graph of implicit function with bifurcation point (Problem on the calculation of the arch)https://en.smath.info/forum/yaf_...thodinEngineering.aspxThe graph consists of three curves that occur at the bifurcation point. 1. Find the coordinates of the bifurcation point 2.The starting point for each of the three graphs is taken near the bifurcation point 3. Build graphics by Draghilev method Point Bifurcation.sm (39kb) downloaded 12 time(s).

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Originally Posted by: Ber7 The graph consists of three curves that occur at the bifurcation point. 1. Find the coordinates of the bifurcation point 2.The starting point for each of the three graphs is taken near the bifurcation point 3. Build graphics by Draghilev method Thanks Ber7. This version works fine compared to the previous "arca" that never stopped pedaling . By same token, I'm puzzled by the Lagrange points. Where those contours come from ?




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Joan from Wikipedia it is stated that they are involved in Astronomy and that for 2 large bodies there are 5 of these points so I guess that your pictures/worksheet refers on how to calculate them all(their positions). https://en.wikipedia.org/wiki/Lagrangian_pointBest regards Franco




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Originally Posted by: Ber7 Thank you, Alvaro. The norme function simplifies the code and reduces the calculation time. Example of using norme when solving ODE dn_GearsBDF is nearly ½ time. I think its Loren tz




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Originally Posted by: алексей_алексей hose who want to try hard and to do better in SMath. Thanks for the suggestion. My head is not oblate like Extraterrestrials !




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Originally Posted by: Razonar . I don't remember to read nothing in the literature about equally spaced points in the numerical solution of the ode, except for the opposite: adaptive steps, but referring for the time variable, not the X,Y,Z solution points. About how you apparently get the same distance between solution points, i.e. sqrt(X^2+Y^2+Z^2), I guess that the background theory must to be in the Draghilev method and how the solution (X,Y,Z) is obtained from the differential equation. You have a very interesting point for investigate and publish about it. Apparently the distance between the points is 1 (guess can be easily proved because you divide by the norme the system), and this seems to provide more stable numerical solutions for the system (can be applied here Lyapunov's theorems?)
Best regards. Alvaro. An article about the effectiveness of the solution of the system diff. equations for parameterization integral curve through arc length Russiy.pdf (608kb) downloaded 18 time(s).

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on 06/01/2019(UTC), on 07/01/2019(UTC)


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Originally Posted by: Ber7 Originally Posted by: Razonar . I don't remember ... An article about the effectiveness of the solution of the system diff. equations for parameterization integral curve through arc length Russiy.pdf (608kb) downloaded 18 time(s). Thanks for the paper. Now you found the keywords for this point, which seems to be "Arc Length Method". This give only 79 results at google search: Results are related with mechanical engineering for finite elements analysis. From the first result, having this attached file: https://scholar.harvard....sios/files/ArcLength.pdf But actually I don't find any appointment nor observation that solution points are equally spaced. Notice that It could be some "obvious" point for, given f(t,x,x' ) = 0, plot for x(t) it's equally spaced if one transform it to f(s,x,x' ) with s as the arc length. But it's immediate for me that the plot for the state space (x,x' ) seems to be equally spaced too, as in some smath examples here in this topic. Also, for the observation that can apply Lyapunov theorems about stability of solutions, there are some references in the 79 google's search results. The application here of Lyapunov is related about the stability of the found solutions, because authors using the parametrization along the arc length for ill conditioned systems. Originally Posted by: Razonar . Apparently the distance between the points is 1 (guess can be easily proved because you divide by the norme the system), and this seems to provide more stable numerical solutions for the system (can be applied here Lyapunov's theorems?)
Best regards. Alvaro. Edited by user 09 January 2019 00:15:12(UTC)
 Reason: Not specified

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